Question

A packaging company has been offered a contract to create gift boxes for handbags. The company needs to buy a special machine to make the boxes the machine costs $3000 and each box costs$2 for labor and materials the handbag maker has agreed to buy each box for $5
Under what conditions should the packaging company accept the contract? How many boxes does the handbag maker need to order for the packaging company to break even? How many boxes does the handbag maker need to order for the packaging company to make money?

Answer 1

To determine the conditions under which the packaging company should accept the contract, we need to analyze costs, revenues, and the break-even point.

### Costs:
1. **Fixed Costs**: The cost of the machine, which is $3000.
2. **Variable Costs per Box**: The cost for labor and materials for each box, which is $2.

### Revenue:
- The selling price per box to the handbag maker is $5.

### Profit Calculation:
To find profit, we can use the following formula:

\[
\text{Profit} = \text{Total Revenue} - \text{Total Costs}
\]

Where:
- Total Revenue = Selling Price per Box × Number of Boxes Sold
- Total Costs = Fixed Costs + (Variable Cost per Box × Number of Boxes Sold)

Let \( x \) be the number of boxes sold.

#### Total Revenue:
\[
\text{Total Revenue} = 5x
\]

#### Total Costs:
\[
\text{Total Costs} = 3000 + 2x
\]

### Break-even Point:
Set Profit to 0 for the break-even point:

\[
0 = 5x - (3000 + 2x)
\]

Simplifying gives:

\[
5x = 3000 + 2x
\]
\[
5x - 2x = 3000
\]
\[
3x = 3000
\]
\[
x = \frac{3000}{3} = 1000
\]

The packaging company needs to sell **1000 boxes** to break even.

### Making Profit:
To make a profit, we need the revenue to exceed the costs:

\[
5x > 3000 + 2x
\]

This simplifies to:

\[
5x - 2x > 3000
\]
\[
3x > 3000
\]
\[
x > \frac{3000}{3} = 1000
\]

Thus, to generate profit, the handbag maker needs to order **more than 1000 boxes**.

### Summary:
- The packaging company should accept the contract if they believe they can sell more than 1000 boxes.
- The company will break even when they sell **1000 boxes**.
- The company will start making a profit when they sell **more than 1000 boxes**.